(14c) Data-Driven Distributionally Robust Control Using Optimal Transport for Gaussian Mixture Models

In real control problems, addressing systems affected by random disturbances is critical. These disturbances introduce inherent uncertainties into system dynamics, necessitating control policies that navigate the trade-offs between desired outcomes and the unpredictable nature of their operational environments [1]. Traditional deterministic optimal control often falls short in real-world applications because of its reliance on precise knowledge of system parameters, struggling to handle uncertainty in control problems. In contrast, stochastic optimal control and robust control are tailored for control problems under uncertainty. Both stochastic optimal control and robust control have been applied in various fields such as finance, engineering, and operations research, where making robust and adaptive decisions is important [2]. However, both stochastic optimal control and robust control come with limitations. For instance, traditional stochastic optimal control, assuming known probability distributions for uncertainties [3], faces challenges when dealing with real-world systems characterized by poorly defined uncertainties. As to robust control, it aims to mitigate worst-case uncertainties, which can result in overly conservative actions if the uncertainty set is not appropriately defined, a task that often proves challenging [4]. Since the gap between assumed and actual distributions can lead to suboptimal or even infeasible control actions, as a compromise, the data-driven distributionally robust control [5] is proposed, which does not need the exact uncertainty distribution or uncertainty set, but only relies on history data of uncertainty realizations.

In this work, we propose a new data-driven distributionally robust control strategy through optimal transport between Gaussian mixture models (GMMs). The method does not depend on pre-defined uncertainty sets or distribution information. Instead, the proposed approach utilizes a GMM based on the available data to construct an ambiguity set for distributionally robust optimization [6], and it obtains the optimal control action through distributionally robust optimization. CVaR approximation is employed to enable a tractable optimization formulation for this distributionally robust optimal control approach. Compared to the traditional stochastic control or robust control methods, the proposed method is more general in accommodating uncertainties within control systems, particularly those exhibiting multi-modal distributions which pose significant challenges in real-world systems. By leveraging the versatility of GMMs as universal approximators for diverse uncertainty distributions, our proposed method bridges this gap, enhancing robustness against a wide range of uncertainties without the need for precise prior knowledge of uncertainty distributions.

Simulation studies are conducted to demonstrate its efficacy. In the simulation studies, we demonstrate the method's versatility in controlling systems under general nonlinear joint chance constraints [7], showcasing its potential in enhancing uncertainty management across complex systems. Through this, we contribute to the advancement of distributionally robust control strategies, offering a more reliable solution to managing uncertainties in control problems.

References

[1] S.-B. Yang, Z. Li, and J. Moreira, "A recurrent neural network-based approach for joint chance constrained stochastic optimal control," Journal of Process Control, vol. 116, pp. 209-220, 2022.

[2] G. Fabbri, F. Gozzi, and A. Swiech, "Stochastic optimal control in infinite dimension," Probability and Stochastic Modelling. Springer, 2017.

[3] K. M. Yenkie and U. Diwekar, "Stochastic optimal control of seeded batch crystallizer applying the ito process," Industrial & Engineering Chemistry Research, vol. 52, no. 1, pp. 108-122, 2013.

[4] C. Shang, X. Huang, and F. You, "Data-driven robust optimization based on kernel learning," Computers & Chemical Engineering, vol. 106, pp. 464-479, 2017.

[5] J. Coulson, J. Lygeros, and F. Dörfler, "Distributionally robust chance constrained data-enabled predictive control," IEEE Transactions on Automatic Control, vol. 67, no. 7, pp. 3289-3304, 2021.

[6] S. Kammammettu, S.-B. Yang, and Z. Li, "Distributionally robust optimization using optimal transport for Gaussian mixture models," Optimization and Engineering, pp. 1-26, 2023.

[7] S.-B. Yang and Z. Li, "Kernel distributionally robust chance-constrained process optimization," Computers & Chemical Engineering, vol. 165, p. 107953, 2022.